Genus and crosscap of solvable conjugacy class graphs of finite groups

The solvable conjugacy class graph of a finite group G, denoted by Γ_sc(G), is a simple undirected graph whose vertices are the non-trivial conjugacy classes of G and two distinct conjugacy classes C, D are adjacent if there exist x∈C and y∈D such that ⟨x,y⟩ is solvable. In this paper, we discuss certain properties of the genus and crosscap of Γ_sc(G) for the groups D_2n, Q_4n, S_n, A_n, and PSL(2,2^d). In particular, we determine all positive integers n such that their solvable conjugacy class graphs are planar, toroidal, double-toroidal, or triple-toroidal. We shall also obtain a lower bound for the genus of Γ_sc(G) in terms of the order of the center and number of conjugacy classes for certain groups. As a consequence, we shall derive a relation between the genus of Γ_sc(G) and the commuting probability of certain finite non-solvable group.